This is the original problem that I need to simplify to solve for and for |f(x) - M| and |g(x) - N| respectively.

Take note of those absolute value signs that should be there.

To deal with this problem, you'll certainly need to know that N is not zero. Given that that is the case, suppose that you choose to be less than . By the triangle inequality, that will imply that g(x) lies between and . That should enable you to make some progress.